By James G. Simmonds
In this article which progressively develops the instruments for formulating and manipulating the sector equations of Continuum Mechanics, the maths of tensor research is brought in 4, well-separated phases, and the actual interpretation and alertness of vectors and tensors are under pressure all through. This new version includes extra workouts. moreover, the writer has appended a bit on Differential Geometry.
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Extra info for A Brief on Tensor Analysis
52) are called the roof (or contravariant) components of v. 31] The gi are functions of the general coordinates which, along the trajectory, are (unknown) functions of time. 58). , gi,j is symmetric in the = The are called the Christoffel indices i andj. This implies that rt rt. symbols of the uj-coordinate system. rt 9 Recall that two dummy indicies on the same level (on the roof in this case) are not to be summed over. iRk. (3. 64) We now throw a move that is typical in tensor analysis: we change the dummy index in the first term on the right from i to k.
32), we have g. = vr(r~~ + r~go) + vO(r~o~ + r~ogo) II I · = gi I go = Vr(r~8~ + r~8g0) + v O(f8og. + n8g8) I I I "rkij~. 31), and collect coefficients of gr and g8, to obtain 8 Note that g,.. = X,T. Or = g•. i o 51 The Christoffel Symbols a = [v r + (r::"v r + r~8V8)Vr + (r~8Vr + n8 V8)V 8]gr I ! + [V 8 + (r~rVr + r~8V8)Vr + (r~8Vr + fZ8V8)V8]g81 == argr + a 8g8. 35) We call a r and a 8 the roof components of a in the (r,lJ) coordinate system. Things are not really as complicated as they appear because a number of the Christoffel symbols vanish.
Vi = gik Vk T:j = gikTkj' Tij = gikTkJ, Tij = gikT~j, etc. gi As indicated, the effect of multiplying a component of a vector or tensor by g ik and summing on k is to raise or lower an index. Thus, for example, the index on Vk is raised, becoming an i, by multiplying Vk by gik. 10. Noting that [g,J = GTG, show that (a). det [giJ = J2 (b). gikgkj = 8]. 11. Show that (a). gi x gj = Eijkgk (b). gk = V2E ijk gi X gj. 12. Ifu = Uigi = Uig i , find formulas for the four different components of the 2nd order tensor u x .